Ramanujan sums and rectangular power sums

For a fixed nonnegative integer u and positive integer n, we investigate the symmetric function ∑d|n(c_d([Formula Presented]))^up_d[Formula Presented], where p_n denotes the nth power sum symmetric function, and c_d(r) is a Ramanujan sum, equal to the sum of the rth powers of all the primitive dth roots of unity. We establish the Schur positivity of these functions for u=0 and u=1, showing that, in each case, the associated representation of the symmetric group S_n decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case u=2.